To model the points in the table with an exponential function of the form \( y = a(b)^x \), we first need to identify a and b using the given values of x and y.
From the table:
- When \( x = 0 \), \( y = -3 \)
- When \( x = 1 \), \( y = -9 \)
- When \( x = 2 \), \( y = -27 \)
- When \( x = 3 \), \( y = -81 \)
Step 1: Find 'a'.
Using the first point \( (0, -3) \): \[ y = a(b)^0 \implies y = a \implies -3 = a \] So, \( a = -3 \).
Step 2: Find 'b'.
Now we can use the second point \( (1, -9) \) to find b: \[ y = -3(b)^1 \implies -9 = -3b \] Dividing both sides by \(-3\): \[ b = \frac{-9}{-3} = 3 \]
Step 3: Verify with other points.
To ensure this is consistent, we can check other points:
-
For \( x = 2 \): \[ y = -3(3)^2 = -3(9) = -27 \quad \text{(Correct)} \]
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For \( x = 3 \): \[ y = -3(3)^3 = -3(27) = -81 \quad \text{(Correct)} \]
Conclusion
The exponential function that models the points in the table is: \[ y = -3(3)^x \]